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Properties

Label 2-1476-164.163-c0-0-1
Degree $2$
Conductor $1476$
Sign $1$
Analytic cond. $0.736619$
Root an. cond. $0.858265$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.41·7-s + 8-s + 1.41·11-s − 1.41·14-s + 16-s + 1.41·19-s + 1.41·22-s − 25-s − 1.41·28-s + 32-s + 1.41·38-s − 41-s + 1.41·44-s − 1.41·47-s + 1.00·49-s − 50-s − 1.41·56-s − 2·61-s + 64-s − 1.41·67-s − 1.41·71-s + 1.41·76-s − 2.00·77-s + 1.41·79-s − 82-s + ⋯
L(s)  = 1  + 2-s + 4-s − 1.41·7-s + 8-s + 1.41·11-s − 1.41·14-s + 16-s + 1.41·19-s + 1.41·22-s − 25-s − 1.41·28-s + 32-s + 1.41·38-s − 41-s + 1.41·44-s − 1.41·47-s + 1.00·49-s − 50-s − 1.41·56-s − 2·61-s + 64-s − 1.41·67-s − 1.41·71-s + 1.41·76-s − 2.00·77-s + 1.41·79-s − 82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1476\)    =    \(2^{2} \cdot 3^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(0.736619\)
Root analytic conductor: \(0.858265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1476} (163, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1476,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.961128310\)
\(L(\frac12)\) \(\approx\) \(1.961128310\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
41 \( 1 + T \)
good5 \( 1 + T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.41T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.705968290597660400750604545000, −9.136129680727659168291888519768, −7.82329451436062946474700126531, −6.97002368675921360142881872629, −6.35259995173532438525362902785, −5.69905878489709890040044807750, −4.55830788875088394314565672338, −3.57154273321310064218252119978, −3.08045646205651256833887920390, −1.57076904818596277278193059180, 1.57076904818596277278193059180, 3.08045646205651256833887920390, 3.57154273321310064218252119978, 4.55830788875088394314565672338, 5.69905878489709890040044807750, 6.35259995173532438525362902785, 6.97002368675921360142881872629, 7.82329451436062946474700126531, 9.136129680727659168291888519768, 9.705968290597660400750604545000

Graph of the $Z$-function along the critical line